\section{Quantitative Analysis Of Insurer Operations}
\label{sec:QuantitativeAnalysisofInsurerOperations}

Insurers $NHI$, $B$, $PI$, $D$ and $E$ randomly select 308,000,000; 10,000,000; 1,000,000; 100,000; and 10,000 policyholders from the same population (See Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 1). Each insurer produces a Population Loss Ratio Estimate (PLRE) of the Population Loss Ratio. How accurate these PLREs will be is measured by each insurer's unique standard error. Insurers select very large, random samples, so I can safely assume that their PLRE probability distribution functions are normally distributed, even when individual policyholder PLREs are not. I will show that transferring insurance risks to health care providers (Capitation) is flawed, by comparing the impact of portfolio size on PLRE variability, PLRE probabilities, and these five insurers' probabilities of earning profits, incurring operating losses, or becoming insolvent. I will also compare insurers' surplus requirements and maximum sustainable benefits for policyholders.

\subsection{Insurer Standard Errors By Portfolio Size}
\label{sec:InsurerStandardErrorsbyPortfolioSize}

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 1, shows insurer portfolio sizes in thousands (1,000s) of policyholders. Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 2, shows portfolio size adjusted standard errors, $\sigma_{e_{N}}$ = $\sigma_{e_{PI}}$ * $\frac{\sqrt{1,000,000}}{\sqrt{N}}$. $NHI$'s standard error, $\sigma_{e_{NHI}}$, is 0.00285 while $\sigma_{e_{E}}$ = 0.50000, ten times larger than $\sigma_{e_{PI}}$ (0.05000), and 175 times larger than $\sigma_{e_{NHI}}$. Tables~\ref{tab:InsurerPremiumsStandardErrorsLossRatioProbabilitiesbyPortfolioPortfolioSize0.01Increment} through \ref{tab:RiskLoadedReinsurancePremiumsByPortfolioSize} show standard errors by portfolio size, for insurers with portfolios varying from a single policyholder to 309,000,000 policyholders, and how insurers' standard errors can be used to analyze many different aspects of insurer performance.

These insurers' normally distributed Population Loss Ratio Estimate Distribution Functions, are: $\Phi_{NHI}$(0.7500, 0.002844); $\Phi_{B}$(0.7500, 0.015811); $\Phi_{PI}$(0.7500, 0.050000); $\Phi_{D}$(0.7500, 0.158114); and $\Phi_{E}$(0.7500, 0.500000).

Capitation advocates appear to have failed to address these profound differences in PLRE Distribution Functions because  \emph{all insurers larger than $PI$ have more probability below PLR + $\epsilon$} ($\epsilon > 0$) than $PI$, and \emph{all smaller insurers have more probability above PLR + $\epsilon$} than $PI$. When correctly analyzed, these distribution function differences cause dramatically different insurer operating results in efficient insurance markets and efficient health care (finance) systems.

% \begin{table}%[bbb] 
% \begin{center}
% \caption{Cumulative PLRE Distribution Functions by Portfolio Size} 
% \begin{tabular}{|crcc|} 
%  \hline
%   Insurer & \multicolumn{1}{c}{Size} &  Standard Error & Cumulative PLRE Distribution Function \\
% \hline
%   NHI & 308,000,000 	& 0.002849 & $\Phi$(0.7500, 0.002849) \\
%   B & 10,000,000 	& 0.015811 & $\Phi$(0.7500, 0.015811) \\
%   PI & 1,000,000  	& 0.050000 & $\Phi$(0.7500, 0.050000) \\
%   D & 100,000  		& 0.158114 & $\Phi$(0.7500, 0.158114) \\
%   E & 10,000  		& 0.500000 & $\Phi$(0.7500, 0.500000) \\
% \hline
% \end{tabular} \label{tab:CDF}
% %\caption{Cumulative Distribution Functions by Portfolio Size} 
% 
% \end{center}
% \end{table}


\subsection{Insurer Profit Probabilities By Portfolio Size}
\label{sec:InsurerProbabilitiesofProfitsbyPortfolioSize}

Insurers use the 85\% of their premiums not allocated to operating expenses in Formula~\ref{eq:PremiumAllocationByCostsProspective} to pay policyholders' health expenses, converting all unused portions to profits. Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 3, shows that all insurers have probability, $\Phi_N(0.7500)$ = 0.5000, of profits of at least 10\%, at PLREs at, or below, the Population Loss ratio (0.7500), because E[$PLRE_N$] = PLR for all insurers. Capitation advocates may not have gone any further than this because this is the only PLRE at which insurers' profit probabilities are identical. Small insurers have more probability in the tails of their distributions which explains why they tend to have volatile operating results. Large insurers tend to have most of their probability close to the PLR, so their operating results tend to be very stable. 

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 4, shows insurers' probabilities of profits of at least 5\%, ($\Phi_N(0.8000)$), at PLREs below 0.8000. $NHI$ earns such profits with probability 1.0000, $B$ with probability 0.9992, and $PI$ with probability 0.8413. $D$ and $E$ have much lower probabilities of earning such profits, 0.6241 and 0.5398, respectively. However, $\Phi_{NHI}(0.8000)$) = 1.0000 is very misleading. $NHI$'s probability, $\Phi_{NHI}$(PLR + 3 * $\sigma_{e_{NHI}}$) = $\Phi_{NHI}(0.758547)$ = 0.9987. $NHI$ \emph{almost always} earns profits greater than 9.13\%, and since $\Phi_{B}(0.79743)$ = 0.9987, $B$ \emph{almost always} earns profits greater than 5.25\%! 

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 5, shows insurers' probabilities of profits greater than 0\% (Break Even), $\Phi_N(0.8500)$, at PLREs below 0.8500. $NHI$ and $B$ have probability 1.0000, $PI$'s break even probability is 0.9772, but $D$ and $E$ have much lower ``break-even`` probabilities, 0.7365 and 0.5793, respectively.

\subsection{Insurer Probabilities Of Operating Losses By Portfolio Size}
\label{sec:InsurerProbabilitiesofOperatingLossesbyPortfolioSize}

$NHI$ and $B$ have probability 0.0000 of incurring operating losses at Population Loss Ratio Estimates in excess of 0.8500. $PI$ has probability 0.0228 of incurring operating losses when its Population Loss Ratio Estimate exceeds 0.8500. Insurers $D$ and $E$ have much higher probabilities of incurring operating losses when their Population Loss Ratio Estimates exceed 0.8500: 0.2635 (1.0000 - 0.7365) and 0.4207 (1.0000 - 0.5793), respectively. 

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 6, shows insurers' probabilities of incurring operating losses greater than 5\% ($1.0000 - \Phi_N(0.9000)$), at PLREs above 0.9000. $NHI$ and $B$ incur such operating losses with probability 0.0000, $PI$'s probability is 0.00135, but $D$ and $E$ have probabilities, 0.1714 and 0.3821, respectively. Insurer D is 127 times more likely, and Insurer E is 283 times more likely, to incur 5\%, or higher, operating losses, than $PI$. 

Table~\ref{tab:InsurerOperatingResultsByPortfolioSize} Row 7, shows insurers' probabilities of operating losses greater than 10\%, at PLREs above 0.9500. $\Phi_{NHI}$(0.9500) = $\Phi_{B}$(0.9500) = $\Phi_{PI}(0.9500)$ = 0.0000, while $\Phi_{D}(0.9500)$ = 0.1030 and $\Phi_{E}(0.9500)$ = 0.3446. $D$ can expect 10\%, or higher, operating losses more than one year in ten, and $E$ more than one year in three.
